Question

simplify
square root of (2square root of 5 - square root of 10)^2

Answer 1

To simplify the expression \(\sqrt{(2\sqrt{5} - \sqrt{10})^2}\), we can use the property of square roots:

\[
\sqrt{x^2} = |x|
\]

Thus, we have:

\[
\sqrt{(2\sqrt{5} - \sqrt{10})^2} = |2\sqrt{5} - \sqrt{10}|
\]

Next, let’s simplify the expression inside the absolute value:

1. We know that \(\sqrt{10} = \sqrt{2 \cdot 5} = \sqrt{2} \cdot \sqrt{5}\), so we can rewrite it:

\[
2\sqrt{5} - \sqrt{10} = 2\sqrt{5} - \sqrt{2}\sqrt{5}
\]

2. Factor out \(\sqrt{5}\):

\[
= \sqrt{5} (2 - \sqrt{2})
\]

So, we have:

\[
|2\sqrt{5} - \sqrt{10}| = |\sqrt{5}(2 - \sqrt{2})|
\]

Since \(\sqrt{5}\) is positive, we can write:

\[
|\sqrt{5}(2 - \sqrt{2})| = \sqrt{5} |2 - \sqrt{2}|
\]

Next, we need to evaluate \(2 - \sqrt{2}\). Since \(\sqrt{2} \approx 1.414\), we have:

\[
2 - \sqrt{2} \approx 0.586 \quad (\text{which is positive})
\]

So the absolute value does not affect the expression. Therefore, we have:

\[
|2 - \sqrt{2}| = 2 - \sqrt{2}
\]

Finally, we can conclude that:

\[
\sqrt{(2\sqrt{5} - \sqrt{10})^2} = \sqrt{5}(2 - \sqrt{2})
\]

Thus, the simplified form is:

\[
\sqrt{5}(2 - \sqrt{2})
\]